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Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model:

$$ Y = K^\beta (AL)^1-\beta $$$$ Y = K^\beta (AL)^{1-\beta} $$

I have been asked to derive the steady state values for capital per effective worker:

$$ k^*=(\frac{s}{n+g+ \delta })^{\frac{1}{1-\beta }} $$$$ k^*=\left(\frac{s}{n+g+ \delta }\right)^{\frac{1}{1-\beta }} $$

As well as the steady state ratio of capital to output (K/Y):

$$ \frac{K^{SS}}{Y^{SS}} = \frac{s}{n+g+\delta } $$

I found both of these fine, but I have been also asked to find the "steady-state value of the marginal product of capital, dY/dK". Here is what I did:

$$ Y = K^\beta (AL)^1-\beta $$$$ Y = K^\beta (AL)^{1-\beta} $$ $$ MPK = \frac{dY}{dK} = \beta K^{\beta -1}(AL)^{1-\beta } $$

Substituting in for K in the steady state (calculated when working out steady state for K/Y ratio above):

$$ K^{SS} = AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }} $$$$ K^{SS} = AL\left(\frac{s}{n+g+\delta }\right)^{\frac{1}{1-\beta }} $$

$$ MPK^{SS} = \beta (AL)^{1-\beta }(AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }})^{\beta -1} $$$$ MPK^{SS} = \beta (AL)^{1-\beta }\left[AL\left(\frac{s}{n+g+\delta }\right)^{\frac{1}{1-\beta }}\right]^{\beta -1} $$

$$ MPK^{SS} = \beta (\frac{s}{n+g+\delta })^{\frac{\beta -1}{1-\beta }} $$$$ MPK^{SS} = \beta \left(\frac{s}{n+g+\delta }\right)^{\frac{\beta -1}{1-\beta }} $$

Firstly I need to know whether this calculation for the steady state value of MPK is correct?

Secondly, I have been asked to Sketch the time paths of the capital-output ratio and the marginal product of capital, for an economy that converges to its balanced growth path "from below".

I am having problems understanding exactly what the balanced growth path is, as opposed to the steady state, and how to use my calculations to figure out what these graphs should look like.

Sorry for the mammoth post, any help is greatly appreciated! Thanks in advance.

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model:

$$ Y = K^\beta (AL)^1-\beta $$

I have been asked to derive the steady state values for capital per effective worker:

$$ k^*=(\frac{s}{n+g+ \delta })^{\frac{1}{1-\beta }} $$

As well as the steady state ratio of capital to output (K/Y):

$$ \frac{K^{SS}}{Y^{SS}} = \frac{s}{n+g+\delta } $$

I found both of these fine, but I have been also asked to find the "steady-state value of the marginal product of capital, dY/dK". Here is what I did:

$$ Y = K^\beta (AL)^1-\beta $$ $$ MPK = \frac{dY}{dK} = \beta K^{\beta -1}(AL)^{1-\beta } $$

Substituting in for K in the steady state (calculated when working out steady state for K/Y ratio above):

$$ K^{SS} = AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }} $$

$$ MPK^{SS} = \beta (AL)^{1-\beta }(AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }})^{\beta -1} $$

$$ MPK^{SS} = \beta (\frac{s}{n+g+\delta })^{\frac{\beta -1}{1-\beta }} $$

Firstly I need to know whether this calculation for the steady state value of MPK is correct?

Secondly, I have been asked to Sketch the time paths of the capital-output ratio and the marginal product of capital, for an economy that converges to its balanced growth path "from below".

I am having problems understanding exactly what the balanced growth path is, as opposed to the steady state, and how to use my calculations to figure out what these graphs should look like.

Sorry for the mammoth post, any help is greatly appreciated! Thanks in advance.

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model:

$$ Y = K^\beta (AL)^{1-\beta} $$

I have been asked to derive the steady state values for capital per effective worker:

$$ k^*=\left(\frac{s}{n+g+ \delta }\right)^{\frac{1}{1-\beta }} $$

As well as the steady state ratio of capital to output (K/Y):

$$ \frac{K^{SS}}{Y^{SS}} = \frac{s}{n+g+\delta } $$

I found both of these fine, but I have been also asked to find the "steady-state value of the marginal product of capital, dY/dK". Here is what I did:

$$ Y = K^\beta (AL)^{1-\beta} $$ $$ MPK = \frac{dY}{dK} = \beta K^{\beta -1}(AL)^{1-\beta } $$

Substituting in for K in the steady state (calculated when working out steady state for K/Y ratio above):

$$ K^{SS} = AL\left(\frac{s}{n+g+\delta }\right)^{\frac{1}{1-\beta }} $$

$$ MPK^{SS} = \beta (AL)^{1-\beta }\left[AL\left(\frac{s}{n+g+\delta }\right)^{\frac{1}{1-\beta }}\right]^{\beta -1} $$

$$ MPK^{SS} = \beta \left(\frac{s}{n+g+\delta }\right)^{\frac{\beta -1}{1-\beta }} $$

Firstly I need to know whether this calculation for the steady state value of MPK is correct?

Secondly, I have been asked to Sketch the time paths of the capital-output ratio and the marginal product of capital, for an economy that converges to its balanced growth path "from below".

I am having problems understanding exactly what the balanced growth path is, as opposed to the steady state, and how to use my calculations to figure out what these graphs should look like.

Sorry for the mammoth post, any help is greatly appreciated! Thanks in advance.

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Solow Model: Steady State v Balanced Growth Path

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model:

$$ Y = K^\beta (AL)^1-\beta $$

I have been asked to derive the steady state values for capital per effective worker:

$$ k^*=(\frac{s}{n+g+ \delta })^{\frac{1}{1-\beta }} $$

As well as the steady state ratio of capital to output (K/Y):

$$ \frac{K^{SS}}{Y^{SS}} = \frac{s}{n+g+\delta } $$

I found both of these fine, but I have been also asked to find the "steady-state value of the marginal product of capital, dY/dK". Here is what I did:

$$ Y = K^\beta (AL)^1-\beta $$ $$ MPK = \frac{dY}{dK} = \beta K^{\beta -1}(AL)^{1-\beta } $$

Substituting in for K in the steady state (calculated when working out steady state for K/Y ratio above):

$$ K^{SS} = AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }} $$

$$ MPK^{SS} = \beta (AL)^{1-\beta }(AL(\frac{s}{n+g+\delta })^{\frac{1}{1-\beta }})^{\beta -1} $$

$$ MPK^{SS} = \beta (\frac{s}{n+g+\delta })^{\frac{\beta -1}{1-\beta }} $$

Firstly I need to know whether this calculation for the steady state value of MPK is correct?

Secondly, I have been asked to Sketch the time paths of the capital-output ratio and the marginal product of capital, for an economy that converges to its balanced growth path "from below".

I am having problems understanding exactly what the balanced growth path is, as opposed to the steady state, and how to use my calculations to figure out what these graphs should look like.

Sorry for the mammoth post, any help is greatly appreciated! Thanks in advance.